The latest measurements of CMB electron scattering optical depth reported by the Planck satellite significantly reduces the allowed range of HI reionization models, pointing towards a later ending and/or less extended phase transition than previously assumed. During HI reionization the intergalactic medium (IGM) is photoheated to ~104 K, and owing to long cooling and dynamical times of the diffuse gas, comparable to the Hubble time, its vestiges persist in the IGM to much later times. Existing and upcoming observations of the Lyman-α (Ly-α) forest at high redshift can be used to detect these extant thermal signatures of HI reionization by comparing them with accurate hydrodynamical models of HI reionization. This can provide new independent constraints not only on the timing of HI reionization but also on the sources that reionized the universe. However to achieve these goals reliable and precise models of the Ly-α forest are needed to be compared with the observations. The challenge for these models is to reach enough dynamical range to attain the necessary precision in the Ly-α forest predictions and at the same time to obtain the number of models required to guarantee a good sampling of the parameter space.